September  2015, 20(7): 2107-2128. doi: 10.3934/dcdsb.2015.20.2107

Existence and uniqueness of steady flows of nonlinear bipolar viscous fluids in a cylinder

1. 

Allstate Insurance Company, 2775 Sanders Road, Suite D2W, Northbrook, IL 60062, United States

2. 

Northern Illinois University, Department of Mathematical Sciences, De Kalb, IL 60115

3. 

Northern Illinois University, Department of Mathematical Sciences, DeKalb, IL 60115-2888, United States

Received  April 2014 Revised  February 2015 Published  July 2015

The existence and uniqueness of solutions to the boundary-value problem for steady Poiseuille flow of an isothermal, incompressible, nonlinear bipolar viscous fluid in a cylinder of arbitrary cross-section is established. Continuous dependence of solutions, in an appropriate norm, is also established with respect to the constitutive parameters of the bipolar fluid model, as these parameters converge to zero, under the additional assumption that the cylinder has a circular cross-section.
Citation: Allen Montz, Hamid Bellout, Frederick Bloom. Existence and uniqueness of steady flows of nonlinear bipolar viscous fluids in a cylinder. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2107-2128. doi: 10.3934/dcdsb.2015.20.2107
References:
[1]

R. A. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar

[2]

G. K. Batchelor, An Introduction to Fluid Dynamics,, Cambridge University Press, (1999). Google Scholar

[3]

H. Bellout and F. Bloom, Incompressible Bipolar and non-Newtonian Viscous Fluid Flow,, Advances in Mathematical Fluid Mechanics, (2014). doi: 10.1007/978-3-319-00891-2. Google Scholar

[4]

H. Bellout, F. Bloom and J. Nečas, Phenomenological behavior of multipolar viscous fluids,, Quarterly of Applied Mathematics, 50 (1992), 559. Google Scholar

[5]

H. Bellout and F. Bloom, Steady plane poiseuille flows of incompressible multipolar fluids,, International Journal of Non-Linear Mechanics, 28 (1993), 503. doi: 10.1016/0020-7462(93)90043-K. Google Scholar

[6]

H. Bellout and F. Bloom, On the uniqueness of plane poiseuille solutions of the equations of incompressible dipolar viscous fluids,, International Journal of Engineering Science, 31 (1993), 1535. doi: 10.1016/0020-7225(93)90030-X. Google Scholar

[7]

H. Bellout and F. Bloom, Existence and asymptotic stability of time-dependent poiseuille flows of isothermal bipolar fluids,, Applicable Analysis, 50 (1993), 115. doi: 10.1080/00036819308840188. Google Scholar

[8]

H. Bellout and F. Bloom, On the higher-order boundary conditions for incompressible nonlinear bipolar fluid flow,, Quarterly of Applied Mathematics, 71 (2013), 773. doi: 10.1090/S0033-569X-2013-01330-9. Google Scholar

[9]

J. L. Bleustein and A. E. Green, Dipolar fluids,, International Journal of Engineering Science, 5 (1967), 323. doi: 10.1016/0020-7225(67)90041-9. Google Scholar

[10]

F. Bloom and W. Hao, Steady flows of nonlinear bipolar viscous fluids between rotating cylinders,, Quarterly of Applied Mathematics, LIII (1995), 143. Google Scholar

[11]

A. Chorin and J. Marsden, A Mathematical Introduction to Fluid Mechanics,, Third Edition, (1993). doi: 10.1007/978-1-4612-0883-9. Google Scholar

[12]

Q. Du and M. Gunzburger, Analysis of a Ladyzhenskaya model for incompressible viscous flow,, Journal of Mathematical Analysis and Applications, 155 (1991), 21. doi: 10.1016/0022-247X(91)90024-T. Google Scholar

[13]

L. C. Evans, Partial Differential Equations,, American Mathematical Society, (1998). doi: 10.1090/gsm/019. Google Scholar

[14]

G. P. Galdi, Mathematical problems in classical and non-Newtonian fluid mechanics,, in Hemodynamical Flows, (2008), 121. doi: 10.1007/978-3-7643-7806-6_3. Google Scholar

[15]

A. E. Green and R. S. Rivlin, Multipolar continuum mechanics,, Archive for Rational Mechanics and Analysis, 17 (1964), 113. Google Scholar

[16]

A. E. Green and R. S. Rivlin, Simple force and stress multipoles,, Archive for Rational Mechanics and Analysis, 16 (1964), 325. Google Scholar

[17]

O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Gordon and Breach, (1969). Google Scholar

[18]

O. Ladyzhenskaya, New equations for the description of the viscous incompressible fluids and solvability in the large of the boundary value problems for them,, in Boundary Value Problems of Mathematical Physics V, (1970). Google Scholar

[19]

J.-L. Lions, Quelques Mèthodes de Rèsolution des Problemes aux Limites Nonlineaires,, Dunod; Gauthier-Villars, (1969). Google Scholar

[20]

A. Montz, Some Bipolar Viscous Fluid Flow Problems in Rigid and Compliant Domains,, Ph.D thesis, (2014). Google Scholar

[21]

J. Nečas, Direct Methods in the Theory of Elliptic Equations,, Springer-Verlag, (2012). doi: 10.1007/978-3-642-10455-8. Google Scholar

[22]

J. Nečas and M. Šilhavý, Multipolar viscous fluids,, Quarterly of Applied Mathematics, 49 (1991), 247. Google Scholar

[23]

Y. R. Ou and S. S. Sritharan, Analysis of regularized Navier-Stokes equations. I, II,, Quarterly of Applied Mathematics, 49 (1991), 651. Google Scholar

[24]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis,, North-Holland Pub. Co., (1977). Google Scholar

[25]

R. A. Toupin, Theories of elasticity with couple-stress,, Archive for Rational Mechanics and Analysis, 17 (1964), 85. Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar

[2]

G. K. Batchelor, An Introduction to Fluid Dynamics,, Cambridge University Press, (1999). Google Scholar

[3]

H. Bellout and F. Bloom, Incompressible Bipolar and non-Newtonian Viscous Fluid Flow,, Advances in Mathematical Fluid Mechanics, (2014). doi: 10.1007/978-3-319-00891-2. Google Scholar

[4]

H. Bellout, F. Bloom and J. Nečas, Phenomenological behavior of multipolar viscous fluids,, Quarterly of Applied Mathematics, 50 (1992), 559. Google Scholar

[5]

H. Bellout and F. Bloom, Steady plane poiseuille flows of incompressible multipolar fluids,, International Journal of Non-Linear Mechanics, 28 (1993), 503. doi: 10.1016/0020-7462(93)90043-K. Google Scholar

[6]

H. Bellout and F. Bloom, On the uniqueness of plane poiseuille solutions of the equations of incompressible dipolar viscous fluids,, International Journal of Engineering Science, 31 (1993), 1535. doi: 10.1016/0020-7225(93)90030-X. Google Scholar

[7]

H. Bellout and F. Bloom, Existence and asymptotic stability of time-dependent poiseuille flows of isothermal bipolar fluids,, Applicable Analysis, 50 (1993), 115. doi: 10.1080/00036819308840188. Google Scholar

[8]

H. Bellout and F. Bloom, On the higher-order boundary conditions for incompressible nonlinear bipolar fluid flow,, Quarterly of Applied Mathematics, 71 (2013), 773. doi: 10.1090/S0033-569X-2013-01330-9. Google Scholar

[9]

J. L. Bleustein and A. E. Green, Dipolar fluids,, International Journal of Engineering Science, 5 (1967), 323. doi: 10.1016/0020-7225(67)90041-9. Google Scholar

[10]

F. Bloom and W. Hao, Steady flows of nonlinear bipolar viscous fluids between rotating cylinders,, Quarterly of Applied Mathematics, LIII (1995), 143. Google Scholar

[11]

A. Chorin and J. Marsden, A Mathematical Introduction to Fluid Mechanics,, Third Edition, (1993). doi: 10.1007/978-1-4612-0883-9. Google Scholar

[12]

Q. Du and M. Gunzburger, Analysis of a Ladyzhenskaya model for incompressible viscous flow,, Journal of Mathematical Analysis and Applications, 155 (1991), 21. doi: 10.1016/0022-247X(91)90024-T. Google Scholar

[13]

L. C. Evans, Partial Differential Equations,, American Mathematical Society, (1998). doi: 10.1090/gsm/019. Google Scholar

[14]

G. P. Galdi, Mathematical problems in classical and non-Newtonian fluid mechanics,, in Hemodynamical Flows, (2008), 121. doi: 10.1007/978-3-7643-7806-6_3. Google Scholar

[15]

A. E. Green and R. S. Rivlin, Multipolar continuum mechanics,, Archive for Rational Mechanics and Analysis, 17 (1964), 113. Google Scholar

[16]

A. E. Green and R. S. Rivlin, Simple force and stress multipoles,, Archive for Rational Mechanics and Analysis, 16 (1964), 325. Google Scholar

[17]

O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Gordon and Breach, (1969). Google Scholar

[18]

O. Ladyzhenskaya, New equations for the description of the viscous incompressible fluids and solvability in the large of the boundary value problems for them,, in Boundary Value Problems of Mathematical Physics V, (1970). Google Scholar

[19]

J.-L. Lions, Quelques Mèthodes de Rèsolution des Problemes aux Limites Nonlineaires,, Dunod; Gauthier-Villars, (1969). Google Scholar

[20]

A. Montz, Some Bipolar Viscous Fluid Flow Problems in Rigid and Compliant Domains,, Ph.D thesis, (2014). Google Scholar

[21]

J. Nečas, Direct Methods in the Theory of Elliptic Equations,, Springer-Verlag, (2012). doi: 10.1007/978-3-642-10455-8. Google Scholar

[22]

J. Nečas and M. Šilhavý, Multipolar viscous fluids,, Quarterly of Applied Mathematics, 49 (1991), 247. Google Scholar

[23]

Y. R. Ou and S. S. Sritharan, Analysis of regularized Navier-Stokes equations. I, II,, Quarterly of Applied Mathematics, 49 (1991), 651. Google Scholar

[24]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis,, North-Holland Pub. Co., (1977). Google Scholar

[25]

R. A. Toupin, Theories of elasticity with couple-stress,, Archive for Rational Mechanics and Analysis, 17 (1964), 85. Google Scholar

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